Advantages

Can be very interactive
Web friendly
No prior knowledge for end users
Makes web page more interactive (feedback forms, games and movies)
Images are seen the same in all Internet browsers
Flash movies have consistent look in all browsers.
Mac / PC / Unix compatible.
Small "footprint" - can load relatively fast with slow internet connection.
Allow you to express your message in a more expressive way.

Flash vs. Animated Images and Java Applets

Animated images and Java applets are often used to create dynamic effects on Web pages.
The advantages of Flash are:

Flash loads much faster than animated images
Flash allows interactivity, animated images do not
Flash does not require programming skills, java applets do